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Abstract
Due to the high design freedom, metasurfaces are widely applied to realize miniaturized optical devices with high performance. Recently, assisted by optical design, monochromatic aberration–corrected lenses, composed of two metasurface layers, have been proposed for a well-focused light spot in the focal plane at large incident angles. However, the focus cannot be tuned in these structures. In this paper, based on the principle of the Moiré lens, a wide-angle ($\pm 30^\circ $) metalens with continuously tunable and monochromatic aberration–corrected focus that consists of two cascaded face-to-face metasurfaces is proposed. The focal length of the lens can be tuned by the mutual rotation of the metasurfaces. Simulation results show that the focal plane is changed to 5.34 µm when the rotation angle is increased by 0.4 rad at the incident wavelength of 810 nm. It is anticipated that the proposed metalens may have a good application prospect in integrated miniaturized imaging systems.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
Metasurfaces, which are ultra-thin two-dimensional structures that consist of metal or dielectric optical subwavelength antennas for wavefront modulation [1], have attracted much research interest in recent years. Because the amplitude, phase, and polarization of the transmitted or reflected wave from one unit antenna are highly dependent on the antenna size, shape, material, spacing, and the surrounding medium, various kinds of devices based on metasurface have been proposed by structure design, such as holograms [2–6], deflectors [7,8] and wave plates [9–11]. Due to the advantages of small size, good integration with semiconductor devices, and high design freedom, metasurfaces are also applied to solve the problem of the traditional geometrical optical elements, such as chromatic and monochromatic aberrations [12]. Solving these aberrations usually requires several lenses in traditional geometrical optical design. However, recent research shows that it is possible to solve the aberrations with only one-layered or two-layered metasurfaces, which dramatically increase the compactness of the optical system. In 2015, Aieta et al. used a grating to eliminate the chromatic aberration of three independent wavelengths [13]. Then a broadband achromatic metalens in almost the entire visible range with double dielectric bar unit cells was proposed [14]. In this design, the unit cells were selected to achieve not only the required phase distribution but also the corresponding group delay. Similarly, Shuming et al. and Wang et al. achieved chromatic aberration corrections using gold and GaN metalenes with square unit cells [15,16]. Besides the chromatic aberration correction, metasurfaces designed for monochromatic aberration correction have also been proposed very recently. In 2016, with the help of optical design, Arbabi et al. used a metasurface doublet structure to correct the monochromatic aberrations of a metalens in the infrared band. The field of view was up to $ \pm {{30}}^\circ $ [17]. Then another monochromatic aberration–corrected metalens with the incident angle of $ \pm 25^\circ $ was realized in the visible [18]. A step zoom metalens with large field-of-view angle ($ \pm 20^\circ $) was also proposed in 2019 [19].
However, all of these metalenses are non-tunable. Although an active metalens realized by mechanically stretching its flexible substrate was proposed [20], more metalenses with flexible focus are highly demanded. In a traditional geometrical optical system, the zoom capability is brought by a lens group that is mechanically movable along the optical axis. This doubtlessly increases the whole size of the system. The question is how to realize a zoomable metalens without the sacrifice in size. The concept of the Moiré lens provides a solution. The Moiré lens was first proposed and experimentally realized by Bernet in 2008 [21,22]. This lens was composed of two parallel phase-plates based on diffractive optical element structure. The phase distribution of the whole lens is the superposition of the two plates and dependent on the mutual rotation angle of the plates. Therefore, by phase construction of the two plates, the focus of the lens is tunable through rotating the plates. Compared with the traditional tuning mechanism, a Moiré lens is superior for its compact size and large tuning range. Recently, a Moiré lens was also designed using a two-layered metasurface structure [23]. However, this Moiré metalens still suffers from aberrations. It is possible to realize more applicable zoomable metalenses by combining the concepts of Moiré structures with aberration-free metalenses.
In this paper, we propose a polarization-independent zoomable metalens corrected for monochromatic aberrations at the wavelength of $\lambda = {{810}}\,\,{\rm{nm}}$. Assisted by optical design and the principle of the Moiré lens, the structures of the metasurfaces are determined. Simulation results show that the focus can be continuously tuned from 9.87 to 15.21 µm, and the lens focuses well at all of the focal planes even when the incident angle is up to $\pm 30^\circ $.
2. DESIGN PRINCIPLE
A. Design of a Metalens Corrected for Monochromatic Aberrations
In the metasurface-based lens design, Eq. (1) is usually used to calculate the required phase of one unit cell [9]:
where $\lambda $ is the wavelength, $f$ is the designed focal length, and $r$ is the distance between one unit cell and the center of the metasurface. Although this equation ensures the waves radiated from the unit cells are in phase at the focus, like traditional spherical lenses, when the incident light has an oblique angle, the transmitted light will have a different focal length or may even not be focused [18], as shown in Fig. 1(a). Using Eq. (1) and Kirchoff’s scalar diffraction theory [12], we calculate the electric field distributions at the focal plane of a metasurface with perfect phase modulation for incident angles of 0°, 10°, 20°, and 30°. The parameters of $f$ and $\lambda $ are set as 15.21 µm and 810 nm. The radius of the metasurface is 14 µm. The results are shown in Fig. 1(b). As shown, with the increased incident angle, the focal effect becomes worse.
Fig. 1. (a) and (c) Schematic illustrations of the focusing effect of a traditional lens and a monochromatic aberration–corrected lens at oblique incident angles, respectively. The blue, green, red, and yellow lines represent $\theta = {{0}}^\circ $, 10°, 20°, and 30°, respectively. $\theta $ is the incident angle. (b) and (d) Theoretically calculated field distributions on the focal plane of corresponding metasurfaces for different incident angles. The radiuses of the two metasurfaces are both 14 µm. Scale bar: 1 µm.
To achieve the function of monochromatic aberrations correction, we use the commercial optical design software Zemax Optic Studio to optimize the geometric profiles of a lens with two even aspheric surfaces at the wavelength of 810 nm. The lens radius is 14 µm, and the material selected is ZnSe, with refractive index of ${n_1} = 2.4675$. After optimization, the geometric profile of the lens as a function of $r$ is calculated by the built-in Zemax equation:
Table 1. Optimized Parameters Calculated by Zemax Optimization Design
B. Design of a Zoomable Moiré Metalens
To achieve the zooming function of the monochromatic aberration–corrected metalens, the Moiré lens structure is used, depicted in Fig. 2(a). It consists of two metasurface layers with silica substrates and amorphous silicon cylindrical nano-pillars and has high refractive index in the near-infrared wavelength range [24]. The radius of each metasurface is 14 µm, and the thickness of the substrates is 1 µm. The incident wavelength is 810 nm, with an incident angle denoted by $\theta $. The distance between the two metasurface layers is denoted by $d$, as shown in the side view of the metalens in Fig. 2(b). Borrowing the concept of the Moiré lens [20], the two metasurfaces have opposite phase distributions ${\varphi _1}$ and ${\varphi _2}$, which are calculated by
Fig. 2. (a) Schematic of the tunable monochromatic aberration–corrected lens, which consists of two face-to-face metasurfaces. The focal plane is tuned by the mutual rotation angle $\beta $ of the two metasurfaces. (b) Side view of the metalens and top view of a single-layer metasurface. $d$ denotes the distance between the two metasurfaces. (c) Simulated transmission and phase of one unit cell, as a function of the nano-pillar radius. The subplot shows the structure of one unit cell.
We designed the lens accordingly. To ensure that the pixels on the two metasurfaces are aligned as much as possible when they are rotated, the nano-pillars are arranged in polar coordinates, as shown in the top view of one metasurface in Fig. 2(b). The geometric parameter sweep of one unit cell is conducted by simulations, using the finite-difference time-domain (FDTD) method (Lumerical Solutions). The schematic diagram of a unit-cell structure is shown in the inset of Fig. 2(c). The cylindrical nano-rods are arranged in a square lattice with a lattice constant of $p = 400\,\,{\rm{nm}}$. The radius and height of the nano-pillars are denoted by ${r_p}$ and $h$, respectively. The refractive indices of ${\rm{Si}}{{\rm{O}}_2}$ and amorphous silicon are 1.45315 and 3.52459339, respectively, at the wavelength of 810 nm [11]. During the sweep, the radius ${r_p}$ is varied from 20 to 115 nm, whereas the height $h$ remains 600 nm. The periodic boundary condition is set at $x$ and $y$ directions, and the perfectly matching layer is set at the $z$ direction (the propagation direction of the plane wave). The obtained transmittance and phase variations of one unit cell with respect to its radius are shown in Fig. 2(c). It is shown that the phase changes over ${{2}}\pi $ in the selected radius range. In addition, the transmittance is generally higher than 0.9, except for a narrow dip in the range of $92\,\,{\rm{nm}} \lt {r_p}\; \lt {{97}}\,\,{\rm{nm}}$. This is caused by the resonance of the incident wave in the a-Si nano-pillar [12], which should be excluded in the design.
Fig. 3. (a) and (b) Theoretically required phase profiles of the two metasurfaces, calculated by Eq. (6), when $\beta = 1\,\,{\rm{rad}}$. (c) Superposition of the two phase-profiles in (a) and (b). (d) and (e) Simulation results of transmitted phase profiles of the two metasurfaces. (f) Simulation results of transmitted phase profile of the bilayer metasurface.
The nano-pillars with proper radius are selected according to the results of Fig. 2(c) and the phases calculated by Eq. (6), which are converted in the region of [0, $2\pi $]. The distance between the two metasurfaces is selected as $d = 100\,\,{\rm{nm}}$ to eliminate the diffraction effect, which may lead to ${\varphi _{\rm{total}}}$ deviating from ${\varphi _1} + {\varphi _2}$. To verify the distance is close enough, we designed two metasurfaces based on Eq. (6) for $\beta = 1\,\,{\rm{rad}}$. Figures 3(a) and 3(b) show the phase distributions of ${\varphi _1}$ and ${\varphi _2}$ that we need. Their superposition ${\varphi _{\rm{total}}}$ is depicted in Fig. 3(c). We calculated the transmitted phase distributions of these two metasurfaces by simulations separately. The results are shown in Figs. 3(d) and 3(e), respectively, which are found to be in agreement with the required phase profiles in Figs. 3(a) and 3(b). We combined the two metasurfaces to form a Moiré structure, as shown in Fig. 2(a), and calculated the transmitted phase distribution of the whole structure by FDTD simulation. The result is shown in Fig. 3(f). By comparison of Figs. 3(c) and 3(f), it is shown that the simulated phase distribution of the bilayer metasurface structure agrees well with the required phase, which indicates the function of monochromatic aberration correction is possible to realize by the structure. The error is brought by the misalignment of the nano-pillars when there is a relative rotation of the two metasurfaces.
3. NUMERICAL SIMULATIONS AND RESULTS
The FDTD simulations are conducted to verify the monochromatic aberration correction and tunability of the proposed metalens. The transmitted field distributions of the metalens, corresponding to $\beta = 1\,\,{\rm{rad}}$, are shown in Fig. 4. Figure 4(a) shows that when the light is normally incident, a focus can be clearly found with the focal plane at $z = 15.21\,\,\unicode{x00B5}{\rm{m}}$, which agrees very well with the designed focal length. The numerical aperture (NA) is 0.68. When the incident angle is increased, Figs. 4(b)–4(d) show that a well-focused light spot can still be found, even when $\theta $ is increased to 30°. The focus is shifted with the incident angle, and the intensity is decreased gradually. Figures 4(e)–4(h) show the field distributions of the focal plane on the $xy$ plane. The obtained field distributions are very similar to those theoretically calculated in Fig. 1(d), which indicate the monochromatic aberration correction effect of the lens.
Fig. 4. Electric field distributions of the metalens focus corresponding to the rotation angle $\beta = 1\,\,{\rm{rad}}$. (a)–(d) Field distributions of the $xz$ plane corresponding to the incident angle $\theta = 0^\circ $, 10°, 20°, and 30°, respectively. The red dotted line indicates the position of the focal plane corresponding to $\theta = 0^\circ $. (e)–(h) Field distributions of the focal plane at $z = 15.21\,\,\unicode{x00B5}{\rm{m}}$, corresponding to the incident angle $\theta = 0^\circ $, 10°, 20°, and 30°, respectively.
We increased the rotation angle to $\beta = 1.2\,\,{\rm{rad}}$. The simulated field distributions of the lens focus, corresponding to various incident angles, are shown in Fig. 5. Figure 5(a) shows that when the incident angle is 0°, the focal length is decreased to 11.55 µm, and the NA becomes 0.77. The focal length is 3.66 µm smaller than that of $\beta = 1$ rad. When $\theta $ is increased, the focal length is increased slightly, as shown by Figs. 5(b)–5(d). However, a desired focus can still be realized on the focal plane, which is demonstrated by Figs. 5(e)–5(h).
Fig. 5. Electric field distributions of the metalens focus corresponding to the rotation angle $\beta = 1.2\,\,{\rm{rad}}$. (a)–(d) Field distributions of the $xz$ plane corresponding to the incident angle $\theta = 0^\circ $, 10°, 20°, and 30°, respectively. The red dotted line indicates the position of the focal plane corresponding to $\theta = 0^\circ $. (e)–(h) Field distributions of the focal plane at $z = 11.55\,\,\unicode{x00B5}{\rm{m}}$, corresponding to the incident angle $\theta = 0^\circ $, 10°, 20°, and 30°, respectively.
We continued to increase the mutual rotation of the two metasurfaces to $\beta = 1.4\,\,{\rm{rad}}$. The field distributions of the focus are shown in Fig. 6. The focal length is changed to 9.896 µm, and the NA is increased to 0.81. Like those of other rotation angles, small focuses are all obtained on the focal plane for different incident angles. Therefore, a tunable and monochromatic aberration–corrected lens is able to be realized by a two-metasurface cascaded structure. The focal length is changed to 5.34 µm ($6.6\lambda $) when the rotation angle is increased by 0.4 rad.
Fig. 6. Electric field distributions of the metalens focus corresponding to the rotation angle $\beta = 1.4\,\,{\rm{rad}}$. (a)–(d) Field distributions of the $xz$ plane corresponding to the incident angle $\theta = 0^\circ $, 10°, 20°, and 30°, respectively. The red dotted line indicates the position of the focal plane corresponding to $\theta = 0^\circ $. (e)–(h) Field distributions of the focal plane at $z = 9.869\,\,\unicode{x00B5}{\rm{m}}$, corresponding to the incident angle $\theta = 0^\circ $, 10°, 20°, and 30°, respectively.
To further study the focusing effect of the proposed lens, the focusing efficiencies corresponding to various NAs and incident angles are calculated based on the simulation results, which are shown in Fig. 7. The focusing efficiency is defined as the ratio of the focused power to the incident power [16]. The maximum focusing efficiency of 16.9% occurs when ${\rm{NA}} = 0.68$ and $\theta = 0^\circ $. When the incident angle or the NA is increased, the focusing efficiency is decreased, which has been demonstrated by other works [17,23]. The minimum focusing efficiency is only 3.4% for ${\rm{NA}} = 0.81$, and $\theta = 30^\circ $. Compared with other reported lenses composed of cascaded metasurfaces [18,19], the focusing efficiency of the proposed metalens is lower, because there is an air gap between the two metasurfaces in our structure to realize the mutual rotation of the two metasurfaces. This causes the light to be reflected and refracted between the two layers, leading to more energy loss. However, when the rotation angle changes, the nano-pillars that should be overlapped on the two metasurfaces will be misaligned, resulting in phase distribution errors and energy loss.
We further studied the focusing effect dependent on the rotation angle $\beta $. For the traditional Moiré lens, when $\beta $ varies from 0 to $\pi $, the focal length will be decreased from infinity to $1/a\lambda $ [21], where $a$ is a constant. However, because the phases of the two metasurfaces are obtained from an even aspheric surface in our design, the focal length as a function of $\beta $ cannot be calculated theoretically. Therefore, we calculate the focal length, and the focusing efficiency varies with $\beta $ by simulations. For simplicity, only efficiencies corresponding to $\theta = 0^\circ $ or 30° are calculated. The results are shown in Fig. 8. As $\beta $ is gradually increased from 0.5 to 2.5 rad, the focal length is decreased from 31.6 to 4.9 µm, and the slope of the focal length curve is decreased with $\beta $, which is similar to that of a traditional Moiré lens. Regardless of whether $\theta = 0^\circ $ or 30°, the focusing efficiency reaches its maximum when $\beta = 1\,\,{\rm{rad}}$, because the output phase distribution for $\beta = 1\,\,{\rm{rad}}$ corresponds to the design results of Zemax. Whether $\beta $ is increased or decreased, the focusing efficiency is decreased. In addition, the focusing efficiency is always the highest for $\theta = 0^\circ $.
Fig. 9. Focusing effect of the lens when $d = 1000\,\,{\rm{nm}}$. (a)–(d) Electric field distributions in the $xz$ plane and the focal plane corresponding to $\beta = 1\,\,{\rm{rad}}$, when $\theta = 0^\circ $ and 30°, respectively. (e)–(h) Electric field distributions in the $xz$ plane and the focal plane corresponding to $\beta = 1.2\,\,{\rm{rad}}$, when $\theta = 0^\circ $ and 30°, respectively. (i)–(l) Electric field distributions in the $xz$ plane and the focal plane corresponding to $\beta = 1.4\,\,{\rm{rad}}$, when $\theta = 0^\circ $ and 30°, respectively. (m) Focusing efficiency of the lens dependent on NA and $\theta $.
Last but not least, the focusing performance dependent on the air gap between the two metasurfaces is studied. In the previous design and simulations, the distance between the two metasurfaces is set as $d = 100\,\,{\rm{nm}}$ to ensure that ${\varphi _{\rm{total}}} = {\varphi _1} + {\varphi _2}$ and avoid errors caused by oblique incidence. However, this is too close to be realized in practical applications. Therefore, it is necessary to conduct simulations for a larger distance. We increased the distance to $d = 1\,\,\unicode{x00B5}{\rm{m}}$, and the simulation results are shown in Fig. 8. The field distributions of the focus in the $xz$ plane and the focal plane, corresponding to various rotation angles and incident angles, are shown in Figs. 9(a)–9(l). For simplicity, only distributions for $\theta = 0^\circ $ and 30° are depicted. It is found that, compared with the studies above, the focal spot becomes larger for all of the situations, which indicates the focusing effect becomes worse for larger distances due to the diffraction effect. However, the obtained focal lengths are the same as those corresponding to $d = 100\,\,{\rm{nm}}$ for all of the incident angles (15.21 µm for $\beta = 1\,\,{\rm{rad}}$, 11.55 µm for $\beta = 1.2\,\,{\rm{rad}}$, and 9.869 µm for $\beta = 1.4\,\,{\rm{rad}}$). This indicates that when $d$ is increased in a relatively small range, the metalens still maintains monochromatic aberration correction and continuous optical zooming characteristics. The focusing efficiencies for the four incident angles and three rotation angles are shown in Fig. 9(m). It is obvious that when $d$ is increased to 1 µm, the focal efficiency is decreased dramatically, especially for large incident angles. This is because for oblique incidence, the phase superposition of the two metasurfaces deviates from the designed phase distributions.
To improve the efficiency, a ${{\rm{SiO}}_2}$ layer with thickness $g$ is added in the air gap between the two metasurface layers to form a Fabry–Perot resonator. The schematic of one unit cell is depicted in Fig. 10(a). By sweeping the thickness of the ${{\rm{SiO}}_2}$ layer in the range from 0 to 1 µm, we calculate the corresponding transmission of the unit cell, which is shown in Fig. 10(b). The highest transmission occurs when $g$ is 900 nm, which is 16.8%. Therefore, a 900 nm thick silica layer is inserted in the Moiré structure of Fig. 9. The field distributions of $\beta = 1\,\,{\rm{rad}}$, corresponding to the incident angle $\theta = 0^\circ $ and 30°, are depicted in Figs. 10(c)–10(f). They are similar to those in Figs. 9(a)–9(d). The focal length stays 15.21 µm, but the intensity of the focus is increased. The focusing efficiencies corresponding to $\theta = 0^\circ $, 10°, 20°, and 30° are shown in Fig. 10(g). The maximum is found to be 7.7%, which is 42.6% higher than that of the structure without the ${{\rm{SiO}}_2}$ layer. This proves that adding a silica layer to the air gap of the Moiré lens improves the efficiency of the system when a small air gap is not guaranteed.
Fig. 10. Enhanced focusing of the Moiré structure with a ${{\rm{SiO}}_2}$ layer between the two metasurfaces. (a) Schematic of one unit cell. (b) Transmission dependent on the thickness of the ${{\rm{SiO}}_2}$ layer. (c)–(f) Field distributions of the structure with a ${{\rm{SiO}}_2}$ layer. (g) Simulated focusing efficiencies corresponding to different incident angles.
4. CONCLUSION
In summary, we present a design method of a polarization-independent metalens, consisting of two cascaded metasurfaces, that can achieve both continuous zooming and monochromatic aberration correction for wide field of view. By optical design, the phase profile by which monochromatic aberrations can be corrected is obtained, and then according to the principle of the Moiré lens, a continuously tunable focus is realized by the mutual rotation of the two metasurfaces and verified by simulations. The focal plane is changed to 5.34 µm ($6.6\lambda $) when the rotation angle is increased by 0.4 rad. It is anticipated that the proposed metalens may have a good application prospect in integrated miniaturized imaging systems in biomedicine, medical imaging, and other fields.
Funding
National Key R&D Program of China (2017YFB1002900); National Natural Science Foundation of China (61875010, 61420106014).
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